Optimal finite-horizon prophet inequality

Determine the optimal prophet constant for the finite-horizon partial-sums stopping problem with full distributional information, thereby resolving whether the existing finite-horizon prophet inequality is tight.

Background

The paper studies stopping rules for the positive partial sums of a finite-horizon random walk and contrasts this setting with the infinite-horizon case. The infinite-horizon partial-sums prophet inequality is described as resolved, whereas the finite-horizon problem lacks the ladder-height decomposition and the associated Markovian renewal structure.

The authors note that the best cited finite-horizon result provides a prophet constant of 0.317, but that its optimality is unknown. This unresolved question concerns the underlying full-information problem and is presented as context for the greater difficulty of the sample-based finite-horizon setting.

References

This is already apparent in the full-information setting. The infinite-horizon setting was resolved by , while the finite-horizon setting remains open, with the best results due to .

— Sample-Based Prophet Inequalities for Random Walks  (2609.26017 - Kleer et al., 22 Sep 2026) in Section 4, Section 4 introductory discussion